Node importance for dynamical process on networks: A multiscalecharacterizationJie Zhang, Xiao-Ke Xu, Ping Li, Kai Zhang, and Michael Small Citation: Chaos 21, 016107 (2011); doi: 10.1063/1.3553644 View online: http://dx.doi.org/10.1063/1.3553644 View Table of Contents: http://chaos.aip.org/resource/1/CHAOEH/v21/i1 Published by the American Institute of Physics. Related ArticlesConedy: A scientific tool to investigate complex network dynamics Chaos 22, 013125 (2012) Multiscale characterization of recurrence-based phase space networks constructed from time series Chaos 22, 013107 (2012) Optimal pinning synchronization on directed complex network Chaos 21, 043131 (2011) Variability of contact process in complex networks Chaos 21, 043130 (2011) Detecting the topologies of complex networks with stochastic perturbations Chaos 21, 043129 (2011) Additional information on ChaosJournal Homepage: http://chaos.aip.org/ Journal Information: http://chaos.aip.org/about/about_the_journal Top downloads: http://chaos.aip.org/features/most_downloaded Information for Authors: http://chaos.aip.org/authors

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Node importance for dynamical process on networks: A multiscalecharacterization

Jie Zhang,1,2 Xiao-Ke Xu,2,3 Ping Li,2 Kai Zhang,4 and Michael Small21Centre for Computational Systems Biology, Fudan University, Shanghai 200433, People’s Republic of China2Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hong Kong,People’s Republic of China3School of Communication and Electronic Engineering, Qingdao Technological University, Qingdao 266520,People’s Republic of China4Life Science Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley,California 94720, USA

(Received 7 October 2010; accepted 19 January 2011; published online 29 March 2011)

Defining the importance of nodes in a complex network has been a fundamental problem in

analyzing the structural organization of a network, as well as the dynamical processes on it.

Traditionally, the measures of node importance usually depend either on the local neighborhood or

global properties of a network. Many real-world networks, however, demonstrate finely detailed

structure at various organization levels, such as hierarchy and modularity. In this paper, we propose

a multiscale node-importance measure that can characterize the importance of the nodes at varying

topological scale. This is achieved by introducing a kernel function whose bandwidth dictates the

ranges of interaction, and meanwhile, by taking into account the interactions from all the paths

a node is involved. We demonstrate that the scale here is closely related to the physical

parameters of the dynamical processes on networks, and that our node-importance measure can

characterize more precisely the node influence under different physical parameters of the

dynamical process. We use epidemic spreading on networks as an example to show that our

multiscale node-importance measure is more effective than other measures. VC 2011 AmericanInstitute of Physics. [doi:10.1063/1.3553644]

Node importance is a basic measure in characterizing the

structure and dynamics of general complex networks.

Real-world networks, however, usually demonstrate finer

and finer structures at smaller spatial scales. To better

understand node importance at different topological

scales of a network, we propose a multiscale measure that

can evaluate the importance of a node at varying scale.

We use epidemic spreading on networks to show that the

scale of this node-importance measure corresponds well

to the physical parameter of epidemic spreading, and our

measure provides a more systematic characterization of

node importance for dynamical processes on networks.

Complex networks, which are composed of a number of

nodes that are interconnected by a set of edges, are widely

observed in a vast range of natural and artificial systems in

recent years, ranging from the brain through the Internet to

human society,1–4 which have been shown to demonstrate

universal features such as small-world5 and scale-free6

effects. Meanwhile, the past decade have also witnessed sig-

nificant advances in which the general concept of complex

networks has spurred many other research areas.7–12

A fundamental problem in analyzing the complex net-

works is to identify the most important nodes or to define the

importance of the nodes,13–15 which is immediately related

to network resilience to attacks and immunization of epidem-

ics. The important nodes usually play a crucial role in the

global organization of the network which, in turn, have sig-

nificant consequences to the dynamical processes taking

place on it, such as synchronization,16–21 epidemic spread-

ing,22,23 navigation,24 random walks,25–27 and so on.28

A variety of centrality measures have been proposed to

determine the relative importance of a vertex within a

graph,15,29–31 particularly for social networks. Examples are

degree centrality (DC, defined as the degree of a vertex),32

betweenness centrality (BC, measures the number of times

that a shortest path travels through the node),33,34 subgraph

centrality35 (SC, characterizes the participation of each node

in all subgraphs in a network), eigenvector centrality36,37

(EVC, defined as the dominant eigenvector of the adjacency

matrix), andcloseness centrality (CC, reciprocal of the sum

of the lengths of the geodesic distance to every other vertex).

Another way to define node importance is the famous

PageRank38,39 (PR). The PageRank of a page (or a node)

depends not only on how many pages point to it but on the

PageRank of these pages as well. PageRank is defined as:

PRi ¼ ð1� dÞ þ dX

j

PRj

kj; (1)

where PRi is the centrality of node i, j runs for all neigh-

bors of i, and kj is the out-degree of node j. The parameter d is

a damping factor between 0 and 1, which is usually set to 0.85.

These measures characterize the node importance from

different angles. DC measures the immediate influence

1054-1500/2011/21(1)/016107/6/$30.00 VC 2011 American Institute of Physics21, 016107-1

CHAOS 21, 016107 (2011)

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among the nodes, rather than long-term effect by counting

the number of immediate neighbors of a node. Compara-

tively, BC and SC take into account more global information

like the number of shortest path and the subgraphs contain-

ing that node. Generally, a good node-importance measure

should include information from different scales, both local

and global. That is, both immediate neighbors and higher

order neighbors of a given node should be considered. More-

over, most real-world networks, either biological or techno-

logical, demonstrate the so-called hierarchical structure,40

that is, more and more detailed structure of the network is pres-

ent at finer topological scales. Considering both the “higher

order” argument and the hierarchical nature of many real-world

networks, we are naturally led to a multiscale characterization

of node importance, which will be introduced next.

In our previous work, we have introduced the kernel

function into networks for community detection.42 A new

framework based on a kernel function has also been devel-

oped to evaluate the correlation pattern emergent in dynami-

cal processes on networks.41 The kernel function K is a

monotonically decreasing function42,43 used to describe the

fact that the strength of interaction between node i and jdiminishes with their shortest path distance xij (see Fig. 1 in

Ref. 41). Directly coupled nodes usually exert a stronger

influence on each other, while this influence tends to weaken

for indirectly linked nodes due to intermediaries. For the

Gaussian kernel, the interaction between node i and j will be

Rði; jÞ ¼ expð�x2ij=2h2Þ, where xij is the shortest path dis-

tance, h is the bandwidth of the kernel which controls its

width, and R is the interaction matrix.

A great advantage of using such kernel function is its

adjustable bandwidth h, which renders spontaneously a mul-

tiscale framework. A narrow kernel indicates that a node will

have impact only on its immediate neighbors; while under a

wide kernel, a node can affect its higher order neighbors and

this order can be conveniently specified by parameter h.

Generally, the importance of a node depends on not only its

immediate neighbors but also the higher order neighbors.

Combining this with the kernel framework, the importance

of node i [which we denote as u(i)] can then be conveniently

defined as the sum of all the influences it sends out (or

receives): uðiÞ ¼PN

j¼1 Rði; jÞ.Here u(i) only involves the influences going through the

shortest path between node i and other nodes. To take into

account other alternative paths between node i and other

nodes (i.e., non-shortest-path route, also called the general-

ized connectivity44), we add another term into u(i)

uðiÞ ¼XN

j¼1

Rði; jÞ þX

p

exp�LðpÞ2

2h2

!; (2)

where p indicates the non-shortest-path routes between node

i and all other nodes, and L(p) is the length of these non-

shortest-path routes. This way our measure u(i) takes into

account the influences from all the paths associated with

node i. The range of interaction is conveniently controlled

by the width h of the kernel function. Under a small h, only

short-range interactions will be counted, while a large hfavors long-range interactions.

Traditional node-importance measures are purely graph-

theoretic, which are seldom associated directly with dynamical

processes on networks. However, the dynamical processes

on networks is an important topic in the study of complex

systems.29 Understanding the relationship between the struc-

ture of complex networked systems and their dynamics (or

function) is expected to shed more lights onto relevant issues

in neuroscience, ecology, sociology, and engineering.45 In

this paper, we will show that u(i) serves as an effective, mul-

tiscale node-importance measure that can evaluate the influ-

ence of a node in dynamical processes on networks under

different physical parameters. Specifically, we use epidemic

spreading on networks as an example.

We consider the susceptible–infected–susceptible (SIS)

model, in which node have two states: susceptible (S) and

infected (I). A susceptible node is infected with rate m at each

time step, and can recover and become again susceptible

with rate d. Epidemic spreading on networks here is a good

example to illustrate the multiscale concept: an infected

node can spread the disease not only to its immediate neigh-

bors but also to its higher order neighbors through the inter-

mediaries. This large scale interaction will be more relevant

when the effective spreading rate k (defined as m=d) is high.

Numerically, the probability that a node transmits the disease

to its nth order neighbor is mn, if the recovery rate d¼ 0.

Under the circumstances that k is relatively large, a hub node

will be persistently infective and can affect even its higher

order neighbors.

To validate the effectiveness of our node-importance

measure u(i) for epidemic spreading on networks, we first

need to find a quantity that originates purely from epidemic

FIG. 1. (Color online) The Zachary karate club network. The top five nodes

(according to degree) are 34, 1, 33, 3, and 2, with their degrees being 17, 16,

12, 10, and 9, respectively. The number of their second-order neighbors are

6, 9, 12, 20, and 13, respectively.

016107-2 Zhang et al. Chaos 21, 016107 (2011)

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spreading (i.e., not graph-theoretic measures) and can reflect

the importance or influence of each node in the spreading of

epidemics. Then we should check the extent to which this

quantity coincides with our measure u(i), which is defined on

a graph-theoretic sense. One quantity that can properly char-

acterize the node importance for epidemic spreading is the

total number of times a node gets infected [denoted by T(i)]during a fixed time interval. If we use 0 and 1 to represent

the S and I state of the nodes, the dynamics of node i is

encoded in a stochastic process, in which T(i) can be conven-

iently computed by counting the transitions from 0 to 1.

The reason why we use the number of infection a node

receives as an importance indicator for epidemic spreading

can be explained from the viewpoint of network immuniza-

tion: the node that are most easily infected [i.e., those with

the largest T(i)] should be segregated or vaccinated in the

first place for effective immunization of the network. There-

fore, the nodes with larger T(i) can readily be taken as more

“influential” or “important.” It should be noted that this

node-importance quantity for epidemic spreading T(i) natu-

rally suggests the need to consider multiscale interactions on

a network, as the infection of node i may come not only

from its immediate neighbors but also from its higher order

neighbors.

In the following text, we will use epidemic spreading on

a simple network, that is, the Zachary karate club network46

(see Fig. 1) to show the effectiveness of our measure u(i).For a given parameter (k¼ m=d), we first run the SIS model

on the network and compute the number of times a node gets

infected (T(i)). Then we derive the node importance u(i)using an exponential kernel function according to Eq. (2).

Since the kernel function has a tunable bandwidth h, we vary

h in a continuous manner until we get a maximum correla-

tion coefficient q between u(i) and T(i). We denote this h as

hmax. Figure 2 shows the correlation coefficient q between

u(i) andT(i) using two different k. As can be seen, q takes a

high value near 0.94, indicating the good correspondence

between the node influence T(i) and our estimation u(i).The correlation coefficient of T(i) with other centrality mea-

sures are q(T,DC)¼ 0.83, q(T,CC)¼ 0.81, q(T,BC)¼ 0.76, and

q(T,EVC)¼ 0.90. The most remarkable feature of our measure

is its multiscale nature. For epidemic spreading on networks,

k determines the range or scale over which a node can exert

influence. When k is low, each node tends to infect only its

immediate neighbors, which is described well by a narrow

kernel (having a small h); when k is relatively large, the

infection can span to higher order neighbors of a node, there-

fore, the epidemic spreading can be better characterized by a

wider kernel function. Using kernel function with different

width h, we can estimate the node importance for epidemic

spreading on networks under various effective spreading rate k.

Now we explicate how and why this multiscale measure

outperforms other node-importance measures. Table I gives

the ranking of the top five nodes in Karate network according

to T(i) (obtained by running SIS model on Karate network),

u(i) (obtained by our kernel method), and other frequently

used measures. As can be seen, when k is small (k¼ 0.3),

each node has a small range of influence, and the most im-

portant one [according to T(i), see the first line in the table]

is node 34, followed by node 1. In this case, T(i) can be well

fitted by a narrow kernel function (hmax¼ 0.66, see the third

line). When k increases to 0.6, however, the most important

node turns out to be node 1 (see the second line). This is

because the nodes now have a wider range of influence and

node 1, which has far more second-order neighbors than

node 34, becomes the most important node. Degree central-

ity, which is a local measure, provides a good prediction of

node importance when k is relatively small (see the fifth line).

Betweenness and closeness centralities, which take into

account global structures, rank node 1 and 34 as the top two,

which is consistent with a larger k. Because these traditional

centrality measures offer no choice of scale, they are not able

to capture the change in node importance at varying scales.

The validity of our measure is further verified on more

general networks, including a scale-free network and an

email network.47 First we generate a scale-free network with

500 nodes and mean degree 3. Then we assortatively mix

this network so that it has a positive assortativity coeffi-

cient48,49 (r¼ 0.60). The reason why we induce assortative

mixing is that the node importance will be dependent on the

degree–degree correlation, and it is, therefore, more complex

and can test different measures effectively. We run the SIS

model on this network (under two different k) and get T(i).As can be seen in Fig. 3, the infection times T(i) at different

k can be well fitted by u(i) with corresponding hmax, indi-

cated by a large correlation coefficient (q near 0.95) between

FIG. 2. (Color online) The correlation coefficient between u(i) and T(i)under k¼ 0.3 and k¼ 0.6, respectively, for Karate network.

TABLE I. Rankings of the nodes (top five) according to SIS simulation (the

first and second lines), our multiscale node-importance measure u(i), and tra-

ditional centrality measures.

Measure Ranking

T (k¼ 0.3) 34 1 33 3 2

T (k¼ 0.6) 1 34 33 3 2

u (h¼ 0.66) 34 1 33 3 2

u (h¼ 1.2) 1 34 3 33 2

DC 34 1 33 3 2

BC 1 34 33 3 32

CC 1 34 33 3 32

EVC 34 1 3 33 2

016107-3 Multiscale node importance measure Chaos 21, 016107 (2011)

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T(i) and u(i). We also calculated the correlation coefficient

of T(i) (k¼ 0.6) with other node-importance measures, and

we found q(T,DC)¼ 0.62, q(T,CC)¼ 0.72, q(T,EVC)¼ 0.52, and

q(T,BC)¼ 0.06, which are much lower than q(T,u) and indicate

the effectiveness our multiscale measure. Next we use a

larger email (with 1133 nodes) network as an example.47

The numerical results are shown in Fig. 4, where we

note that the correlation coefficient between T(i) and uis about 0.94. In comparison, q(T,DC)¼ 0.79, q(T,CC)¼ 0.86,

q(T,EVC)¼ 0.69, q(T,BC)¼ 0.56. These results suggest our

node-importance measure u(i) can capture the influence of

nodes in epidemic spreading under different k more reliably

than other frequently used importance measures.

In the following text, we will show that our multiscale

node-importance measure u(i) incorporates various sources

of information that are usually captured by traditional cen-

trality measures separately. Take the email network as an

example, when the kernel bandwidth h is small, each node is

influencing only its immediate neighbors. In this case, u(i) is

very much like degree, see Fig. 5, and the correlation coeffi-

cient between the degree sequence and u(i) is 0.99. Under a

large kernel bandwidth h (which means that each node is

exerting influence to its higher order neighbors), our measure

takes into account more global information like betweenness

centrality (BC). The correlation coefficient between the BC

and u(i) is 0.80. By choosing an appropriate bandwidth (e.g.,

at mesoscale), our measure is expected to reflect both the

local and global organization of the network.

Finally, we check PageRank in detail, as it also has a

tuning parameter d defined between 0 and 1, which changes

the global behavior of this metric. First, we examine the cor-

relation between PR and node degree as d changes (see Fig. 6),

which we find to reach a maximum at d¼ 1. This indicates

that PR captures more local structure of the network with a

larger d, as degree measures only local influence.

Then we examine the relation between PageRank and

our measure u. We first set d at a high value (d¼ 0.85, which

is commonly adopted in PR). We find that the correlation

coefficient between PR and u reaches maximum at a

small kernel bandwidth h (h¼ 0.9, corresponding roughly to

first-order neighbors), see Fig. 7. This result is consistent

with that in Fig. 6, that is, a high d renders PR more of a

local metric. When d takes a lower value, however, we find

the relation between PR and u to be complicated, which will

be discussed elsewhere. In fact PageRank can be interpreted

in terms of a random walk of a web surfer wandering on the

web. At each step, it either jumps to any other page on the

web randomly, or jumps to a page that is linked to previous

page randomly. The former occurs with probability 1�d, the

latter with probability d. When d is close to 0, all nodes have

the same centrality, PRi¼ 1. Therefore, no information about

the topology is included. On the contrary, when d is close to

1, the centrality of the node will read PRi ¼P

j PRj

�kj. The

surfer can just walk from a node to one of its neighbors, that

is, local information of the network is taken into account. In

this case, PR is the dominant eigenvector of the normalized

Laplacian of the network.

In summary, we have proposed a multiscale node-im-

portance measure for a network, which can characterize the

influence of a node at various organization levels or scales.

This is achieved by introducing the kernel function into net-

works, which quantifies the interaction among the nodes in a

multiscale manner. Since the kernel function is tunable in its

FIG. 3. (Color online) The correlation coefficient between u(i) and T(i)under k¼ 0.3 and k¼ 0.6, respectively, for assortative Barabasi-Albert (BA)

network.

FIG. 4. (Color online) The correlation coefficient between u(i) and T(i)under k¼ 0.3 and k¼ 0.6, respectively, for an email network (Ref. 47).

FIG. 5. (Color online) The relationship between u(i) and traditional centrality

measures. (a) u(i) and degree centrality (b) u(i) and betweenness centrality.

016107-4 Zhang et al. Chaos 21, 016107 (2011)

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bandwidth, we can define a node-importance measure that is

multiscale in nature. We find that our measure can provide a

good estimation of the node influence in terms of epidemic

spreading on networks. In particular, the scale that is

described by kernel bandwidth h corresponds well to the

effective spreading rate k. Usually, in order to evaluate pre-

cisely the node importance associated dynamical processes

(like epidemic spreading on a network) at a given physical

parameter (such as k), we will need a priori knowledge of

the functional relation between this physical parameter (i.e.,

k) and hmax. For general networks that are not related to dy-

namical processes, we can use Eq. (2) directly to obtain the

multiscale node importance by varying kernel bandwidth h.

The complexity of our algorithm is about O(n2). For

each node (node i), the complexity of finding its first-order

neighbor Ni is O(n). The second-order neighbor of node i can

be easily identified from the first-order neighbor of Ni, and

the third-order neighbor of node i can be identified from the

first-order neighbor of the second-order neighbor of node i,and so on. Therefore, for each node, the complexity of find-

ing neighbors (up to higher order) is generally bounded by

O(n). The overall complexity is O(n2). Furthermore, our

multiscale node-importance measure can also be extended to

characterize other important topological properties of a net-

work, such as rich-club organization and degree–degree mix-

ing in a multiscale manner,49,50 where the degree of a node

can be readily substituted with our node importance u(i).

This work is funded by Hong Kong Polytechnic Univer-

sity (G-YX0N). J. Zhang and Xiao-ke Xu also acknowledge

the support from National Natural Science Foundation of

China under Grant No. 61004104.

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